In MR imaging, a k space is usually used to hold data samples acquired from a subject, e.g., a patient. The locations of the data samples in the k space may be one of the concerns during reconstruction of an image of the subject from the data samples. Most conventional methods for data collection collect the data samples uniformly on the Cartesian grid in the k space. This allows performing image reconstruction by means of direct implementation of fast Fourier transform (FFT) algorithms on the data samples in the k space.
However, some methods may collect data samples located on a non-Cartesian grid or located non-uniformly on a Cartesian grid in a k space. In this case, it is necessary to grid such collected data samples uniformly onto the Cartesian grid in the k space. Such a process may be referred to as a gridding process. After the gridding process, the data samples may be subjected to fast Fourier transform (FFT) algorithms to achieve image reconstruction. The gridding process normally consists of several standard steps including density weight estimation, kernel convolution and roll-off correction. The first step, density weight estimation, in the gridding process is to estimate sample density weights for the data samples in the k space to compensate for non-uniformity of the sampling.
There are various methods for estimating the sample density weights, including analytical methods and iterative methods. The analytical methods can be applied only in rare cases. The iterative methods can be applied universally to any sampling patterns in which the data samples are acquired. However, for each iteration, an order of O(N·LM) operations is needed, with N, L, M being a number of samples, a size of a convolution kernel to be used in the gridding process and a dimension of a k-space, respectively. Therefore, time efficiency could be an issue with a huge number of samples or with a large convolution kernel array size. Furthermore, using the iterative methods, the sample density weights normally must be prepared before image reconstruction starts; if the k-space trajectory changes during a scanning time period, the initial sample density weights do not apply and a recalculation of the sample density weights at scan time is prohibitive due to the high computation load.